An identifying code C of a graph G is a dominating set of G such that
any two distinct vertices of G have distinct closed neighbourhoods within
C. The smallest size of an identifying code of G is denoted
γID(G). When every vertex of G also has a neighbour in C,
it is said to be a total dominating identifying code of G, and the smallest
size of a total dominating identifying code of G is denoted by
γtID(G).
Extending similar characterizations for identifying codes from the
literature, we characterize those graphs G of order n with
γtID(G)=n (the only such connected graph is P3) and
γtID(G)=n−1 (such graphs either satisfy
γID(G)=n−1 or are built from certain such graphs by adding a
set of universal vertices, to each of which a private leaf is attached).
Then, using bounds from the literature, we remark that any (open and closed)
twin-free tree of order n has a total dominating identifying code of size at
most 43n. This bound is tight, and we characterize the trees
reaching it. Moreover, by a new proof, we show that this bound actually holds
for the larger class of all twin-free graphs of girth at least 5. The cycle
C8 also attains this bound. We also provide a generalized bound for all
graphs of girth at least 5 (possibly with twins).
Finally, we relate γtID(G) to the related parameter
γID(G) as well as the location-domination number of G and
its variants, providing bounds that are either tight or almost tight